To understand it, you have to have a starting point. With the example from before with Billy and Susie, you start with their common relationship - lets make it Grandpa. They're on the same "level" for lack of a better word. They're both related to Grandpa the same - they're the grandkids. Their kids are a "level" down, being the great-grandkids. Their kids, the great-great grandkids, are another level down.
So when you're on a different "level" from someone, that's where the removal comes in. Billy is SusieQ's first cousin once removed because there's one level (or generation, I guess, now that I think about it) difference. They're not related to Grandpa in the same way - one is a grandkid and another is a great-grandkid.
Therefore, the "twice removed" means they are two levels (generations) difference. Billy (the grandkid) and SusieQ's child (the great great grandkid) are first cousins twice removed. Billy Jr and Susie Q are second cousins - no removal because they are related to Grandpa in the same way - both are great-grandkids.